Gaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures, κ₁ and κ₂, at the given point:

For example, a sphere of radius r has Gaussian curvature 1/r² everywhere, and a flat plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Gaussian curvature is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in any space. This is the content of the Theorema egregium.

Gaussian curvature is named after Carl Friedrich Gauss, who published the Theorema egregium in 1827.

At any point on a surface, we can find a normal vector that is at right angles to the surface; planes containing the normal vector are called normal planes. The intersection of a normal plane and the surface will form a curve called a normal section and the curvature of this curve is the normal curvature. For most points on most surfaces, different normal sections will have different curvatures; the maximum and minimum values of these are called the principal curvatures, call these κ₁, κ₂. The Gaussian curvature is the product of the two principal curvatures Κ = κ₁ κ₂.

The sign of the Gaussian curvature can be used to characterise the surface.

Most surfaces will contain regions of positive Gaussian curvature (elliptical points) and regions of negative Gaussian curvature separated by a curve of points with zero Gaussian curvature called a parabolic line.

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